When is the quotient space of a second countable space second countable?

Let $\sim$ be the following equivalence relation on $\Bbb R$: $x\sim y$ if and only if $x=y$, or $x,y\in\Bbb Z$. In other words, the $\sim$-equivalence classes are $\Bbb Z$ and the singletons $\{x\}$ for $x\in\Bbb R\setminus\Bbb Z$. Let $z$ be the point of the quotient space corresponding to $\Bbb Z$; then the quotient is not first countable at $z$, so it’s not second countable, even though $\Bbb R$ is. Also, the quotient map is not open, since the image of $\left(-\frac12,\frac12\right)$ does not contain a nbhd of $z$.

A map $q:X\to Y$ is a quotient map if it’s surjective, and $U\subseteq Y$ is open in $Y$ if and only if $q^{-1}[U]$ is open in $X$. This condition ensures that $q$ is continuous, but it does not require $q$ to be open (or closed).

Take $\mathbb R$ and identify $\mathbb Z$ to a point. This will give you a quotient space that is a point with countably infinitely many loops coming out of it. A set in the quotient is open if and only if its pullback via the quotient map is open. In particular, any open neighborhood of the central point must contain a point from each one of the loops, for otherwise the pullback will have isolated points and hence not be open.

The quotient map is not open, as for example $(-1/2,1/2)$ does not map to an open set because the image contains the central point but does not include any points from all but 2 of the loops. The quotient space also does not have a countable basis since it is not even first countable: given any countable collection of open sets containing the central point we may, using a diagonal argument, construct an open set containing the central point that does not contain any of them.